Rabbi Binyamin ben Matitya writes (Shu”t Binyamin Ze’ev § 163) that the honors that are given out at the time the Torah is read, such as raising it for the congregation to see or the different Aliyot, should never be a source of discord in the synagogue. He writes that it is the tendency of humble people to relinquish such honors, while those that he describes as “hungry”, try to grab them. He also bases his opinion on the Halacha that one may not answer “Amen” to someone who forcefully imposed oneself to be the Shaliah Tzibbur without the congregation’s approval. Similarly, it is unbecoming for a congregant to seek out one of the Mitzvot associated with the Torah reading. Furthermore, even if one was rightfully entitled to an Aliya, such as one observing a Yahrtzeit, but it was taken by someone else, the former should not quarrel with that person.
One way to avoid such disharmony is the practice of auctioning off these honors, even on Shabbat. The Rama (Orah Haim 306:6) writes that there is an opinion that it is improper to auction off the Mitzvot on Shabbat since it has the appearance of a weekday transaction, and that even if such a sale does take place, the sale amounts should not be mentioned. Nevertheless, the Rama says that that the custom to be lenient in this matter. Indeed, Rabbi Baruch Toledano (Kitzur Shulhan Aruch, Hotza’at Sefer Torah, pg, 125) and Rabbi David Ovadia (Nahagu Ha’am, Hilchot Shabbat § 20) say that since the sale of Mitzvot is for the sake of charity and communal need, it is permitted do so on Shabbat.
Summary: One should not quarrel over the Mitzvot associated with the Sefer Torah. Auctioning off these Mitzvot is permitted, even on Shabbat.